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As a result the corresponding PCTC takes the form: v1 d(µk + iνk) = −4ν0 eQk+1 −Qk − eQk −Qk−1 − v2 ξk − , (42) dt 2 i dQk (1) = −4ν0 (µk + iνk) − iDk − dt N N (1) Dj . (43) j=1 If we now differentiate (43) and make use of (42) we get: d2 Qk dt2 = 16ν02 eQk+1 −Qk − eQk −Qk−1 + 4ν0 v2 ξk + (1) dD i − i k − dt N N j=1 v1 2 (1) dDj dt . (44) It is reasonable to assume that v2 O( 0 /N ); this ensures the possibility to have the N -soliton train ‘inside’ the potential. It also means that 26 NONLINEAR WAVES: CLASSICAL AND QUANTUM ASPECTS (1) both the exponential terms and the correction terms Mk are of the same order of magnitude.

One can also describe analytically the set of soliton parameters responsible for each of the dynamical regimes. We will show how this can be done for the BSR of the two-soliton trains. From the discussion above these sets of parameters are such that Re (ζ1 − ζ2 ) = 0 where the eigenvalues are the roots of the characteristic equation of L0 = L|t=0 : det(L0 − ζ) = ζ 2 − (b1 + b2 )ζ + b1 b2 − a21 = 0. (25) What one should do now is to solve (25) and express ζ1 − ζ2 in terms of bk and a1 . The BSR is obtained provided the discriminant D of (25) is negative: D = (b1 − b2 )2 + 4a21 < 0.

Indeed, we can assume that initially the solitons are ordered in such a way that ξk+1 − ξk r0 . One can check (p) (p) Mk exp(−2ν0 |k −p|r0 ). Therefore the interaction [3, 10] that Nk terms between the k-th and k±1-st solitons will be of the order of e−2ν0 r0 ; the interactions between k-th and k ± 2-nd soliton will of the order of e−2ν0 r0 . e−4ν0 r0 (0) (0) The terms Ξk , Xk are of the order of r0a exp(−2ν0 r0 ), where a = 0 or 1. However they can be neglected as compare to µk and νk, where √ √ µk = µk − µ0 ν k = νk − ν0 (37) 0, 0.

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